Answer:
666 m
Explanation:
You want the length of AB in a triangle with BC=520, CA=292, and A=48.4°.
Law of sines
Given two sides and a non-included angle of a triangle, we can solve the triangle using the Law of Sines. We can find the unknown side when we know its opposite angle. We can determine its opposite angle after we find the angle opposite the other known side.
sin(B)/b = sin(A)/a
B = arcsin(b/a·sin(A)) . . . . . . solve for angle B
B = arcsin(292/520·sin(48.4°)) ≈ 24.829°
Included angle
The angle opposite the unknown side is ...
C = 180° -A -B = 180° -48.4° -24.829°
C ≈ 106.771°
Unknown side
Then the unknown side is ...
c/sin(C) = a/sin(A)
c = a·sin(C)/sin(A) = (520 m)·sin(106.771°)/sin(48.4°)
c ≈ 666 m
The distance from A to B is about 666 meters.
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Additional comment
You may notice that we maintained full precision of intermediate results, rounding only at the end of the calculation.
Interestingly, since sin(x) = sin(180°-x), we could use ...
c = a·sin(A+B)/sin(A)
without ever having to calculate the value of angle C. Doing it this way avoids any error from the calculation of angle C and saves a step in the solution process.
The fact that the given angle (48.4°) is opposite the longest of the given sides (520 m) means the triangle has one unique solution.
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